Question

What is `(-2,9)` in polar coordinates?

Answer 1

`(r,theta)=(sqrt{85},arctan(-9/2)+pi) approx (9.22,1.79)`, where the `1.79` is the angle measure in radians.

Explanation:

The polar coordinates `(r,theta)` of a point in the plane are related to the rectangular coordinates of the point by the equations `r^{2}=x^{2}+y^{2}` and `tan(theta)=y/x` (when `x !=0`).

Since the point whose rectangular coordinates are `(x,y)=(-2,9)` is in the 2nd quadrant of the plane, if we take `r=sqrt{x^{2}+y^{2}}=sqrt{4+81}=\sqrt{85}`, then we need to add `pi` radians to the output of the arctangent function to find the angle: `theta=arctan(y/x)+pi=arctan(9/(-2))+pi=arctan(-9/2)+pi`.